I would say that it is highly unlikely that, for a device drawing up to 1.5A at 12V, an AC adapter capable of delivering only 0.6A at 9V would be sufficient.
Based on the numbers you gave, the WDS-3600 needs up to 1.5A × 12V = 18W of power.
The AC adapter you got with it can deliver a maximum of 0.6A × 9V = 5.4W, or about one third of what the router is stated as requiring at most.
What's more, the adapter is underrated on both current and voltage.
The undervoltage might possibly be acceptable, but such a large difference (25% lower than rated) is fairly likely to lead to unpredictable behavior and erratic errors. To ensure proper operation, the voltage must match; a small error can be tolerable, but 25% is not a small error.
The undercurrent is highly likely to not be acceptable. From 0.6A to 1.5A is quite a major difference. Larger current rating than needed is acceptable (though, from an efficiency standpoint, not necessarily ideal) but less than needed will at best work, but not reliably. At worst, by using an underrated AC adapter, you can burn out the AC adapter, breaking it and (depending on its internal construction) possibly even causing a fire hazard.
I would return the router to the place of purchase, explain what happened and tell them that you want one with the correct AC adapter. My guess is that this is a packaging mistake somewhere along the delivery line.